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STOP Early Journal Content on JSTOR, Free to Anyone in the World This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in the world by JSTOR. Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid-seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non-commercial purposes. Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- journal-content . JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. CRITICISMS AND DISCUSSIONS. 309 To bring these magical squares to the surface the squares of each set of parallel squares may be permuted as follows: Original order 1, 2, 3, 4, 5, 6, Permuted order 3, 2, 1, 6, 5, 4. The result is the final cube shown in the beginning of this article. The above permutation is subject to two conditions. The sev- eral sets of parallel squares must all be permuted in the same man- ner. Any two parallel squares which in the original cube are located on opposite sides of the middle plane of the cube and at an equal distance from it, in the permuted cube must be located on opposite sides of the middle plane of the cube and at an equal distance from it. These conditions are for the protection of the diagonals. John Worthington. MAGIC IN THE FOURTH DIMENSION. Definition of terms : Row is a general term ; rank denotes a hori- zontal right-to-left row; file a row from front to back; and column a vertical row in a cube — not used of any horizontal dimension. If n 2 numbers of a given series can be grouped so as to form a magic square and n such squares be so placed as to constitute a magic cube, why may we not go a step further and group n cubes in relations of the fourth dimension ? In a magic square containing the natural series 1 . . . w 2 the summation is— — ' ; in a magic cube fl( fl^-i— T ^ with the series i...» 3 it is — - — ^— -; and in an analogous fourth- dimension construction it naturally will be — - — — — . With this idea in mind I have made some experiments, and the results are interesting. The analogy with squares and cubes is not perfect, for rows of numbers can be arranged side by side to repre- sent a visible square, squares can be piled one upon another to make a visible cube, but cubes cannot be so combined in drawing as to picture to the eye their higher relations. My expectation a priori was that some connection or relation, probably through some form of diagonal-of-diagonal, would be found to exist between the cubes containing the n* terms of a series. This particular feature did ap- pear in the cases where n was odd. Here is how it worked out: I. When n is odd. 1. Let w=3, then 8=123. — The natural series 1. . .81 was di- vided into three sub-series such that the sum of each would be 3io THE MONIST. one-third the sum of the whole. In dealing with any such series when n is odd there will be n sub-series, each starting with one of the first w numbers, and the difference between successive terms will be w+i, except after a multiple of n, when the difference is i. In the present case the three sub-series begin respectively with i, 2, 3, and the first is 1 5 9 10 14 18 19 23 27 28 32 36 37 41 45 46 50 54 55 59 63 64 68 72 73 yy 81. These numbers were arranged in three squares constituting a magic cube, and the row of squares so formed was flanked on right and left by similar rows formed from the other two sub-series (see Fig. 1). It is not easy — perhaps it is not possible — to make an abso- lutely perfect cube of 3. These are not perfect, yet they have many 25 38 60 28 77 18 67 8 48 33 79 11 72 1 SO 21 40 62 65 6 52 23 45 55 35 75 13 29 78 16 68 9 46 26 39 58 70 2 5i 19 41 63 31 80 12 24 43 56 36 73 14 66 4 53 69 7 47 27 37 59 30 76 17 20 42 61 32 81 10 71 3 49 34 74 15 64 5 54 22 44 57 Fig. 1. (3') striking features. Taking the three cubes separately we find that in each all the "straight" dimensions — rank, file and column — have the proper footing, 123. In the middle cube there are two plane diagonals having the same summation, and in cubes I and III one each. In cube II four cubic diagonals and four diagonals of vertical squares are correct ; I and III each have one cubic diagonal and one vertical-square diagonal. So much for the original cubes; now for some combinations. The three squares on the diagonal running down from left to right will make a magic cube with rank, file, column, cubic diagonals, two plane diagonals and four vertical-square diagonals (37 in all) correct. Two other cubes can be formed by starting with the top squares of II and III respectively and following the "broken diag- CRITICISMS AND DISCUSSIONS. 3" onals" running downward to the right. In each of these S occurs at least 28 times (in 9 ranks, 9 files, 9 columns and one cubic diag- onal). Various other combinations may be found by taking the squares together in horizontal rows and noting how some columns and assorted diagonals have the proper summation, but the most important and significant are those already pointed out. In all the sum 123 occurs over 200 times in this small figure. 317 473 604 10 161 T92 348 479 510 36 67 223 354 385 536 567 98 229 260 4" 442 598 104 135 286 110 ■36 292 448 579 6to n 167 323 454 485 5" 42 198 329 360 386 542 73 204 235 261 4'7 573 79 423 554 85 236 267 298 429 585 ITI 1+2 '73 304 460 61 1 ■ 7 48 179 335 486 517 548 54 210 361 392 21 r 367 398 529 60 86 242 273 404 560 586 "7 148 279 435 461 6'7 23 154 3'0 336 492 523 29 185 504 35 186 342 498 379 535 6i 217 373 254 410 561 92 248 129 285 436 592 '23 4 160 3" 467 623 606 12 168 324 455 481 512 43 '99 330 356 387 543 74 205 231 262 418 574 80 106 137 293 449 580 299 430 58l 112 '43 (74 305 456 612 18 49 180 33' 487 5'8 549 55 206 362 393 424 555 81 237 268 87 243 274 405 556 587 118 149 280 43' 462 618 24 '55 306 337 493 524 30 181 212 368 399 530 56 380 531 62 218 374 255 406 562 93 249 130 281 437 593 124 5 '56 3'2 468 624 505 3' '87 343 499 ■93 349 480 506 37 68 224 355 38> 537 568. 99 230 256 412 443 599 105 131 287 3'8 474 605 6 162 '75 3t" 457 613 '9 50 176 332 488 5'9 550 5' 207 363 394 425 55' 82 238 269 300 426 582 "3 '44 588 l'9 150 276 432 463 619 25 151 307 338 494 525 26 182 213 309 400 526 57 88 244 275 40J 557 25 r 407 563 94 250 126 282 438 594 125 1 , '57 3'3 469 625 501 32 188 344 500 376 532 63 219 375 38 €9 225 351 382 538 569 100 226 257 4U 444 600 101 132 288 319 475 601 7 ■63 '94 350 476 507 482 5J3 44 200 326 357 388 544 75 201 232 263 419 575 76 107 138 294 450 576 607 13 169 325 45' 464 620 21 rS2 308 339 495 521 27 183 214 370 396 527 58 89 245 271 402 558 589 120 146 277 433 127 283 439 595 121 2 158 J'4 470 621 502 33 189 345 496 377 533 64 220 37' 252 408 564 95 246 570 96 227 258 414 445 596 102 '33 289 320 47' 602 8 164 '95 346 477 508 39 7P 221 352 383 539 358 389 545 71 202 233 264 420 57' 77 108 '39 295 446 577 608 '4 '70 321 452 483 5'4 45 196 327 46 177 333 489 520 546 52 208 364 395 421 552 83 239 270 296 427 583 "4 '45 171 302 458 614 20 3 159 3J5 466 622 503 34 190 34' 497 378 534 65 216 372 253 409 565 9' 247 128 284 440 59' 122 441 597 '03 134 290 3.6 472 603 9 '65 191 347 478 509 40 66 222 353 384 540 566 97 288 259 4'5 234 265 416 572 78 109 140 291 447 578 609 '5 166 322 453 484 5'5 4' 197 328 359 390 54' 72 203 547 53 209 365 391 422 553 84 240 266 297 428 584 "5 14' 172 303 459 615 16 47 ■ 78 334 490 516 340 491 5« 28 ■84 215 366 397 528 59 90 241 272 403 559 590 116 '47 278 434 465 616 22 '53 30S Fig. 2. (s 4 ) One most interesting fact remains to be noticed. While the three cubes were constructed separately and independently the figure formed by combining them is an absolutely perfect square of 9, with a summation of 369 in rank, file and corner diagonal (besides all "broken" diagonals running downward to the right), and a perfect 312 THE MONIST. balancing of complementary numbers about the center. Any such pair, taken with the central number 41, gives us the familiar sum 123, and this serves to bind the whole together in a remarkable manner. 2. Let w=5, then 8=1565. — In Fig. 2 is represented a group of 5-cubes each made up of the numbers in a sub-series of the nat- ural series 1 . . .625. In accordance with the principle stated in a pre- vious paragraph the central sub-series is 1 7 13 19 25 26 32 ... 625, and the other four can easily be discovered by inspection. Each of the twenty-five small squares has the summation 1565 in rank, file, corner diagonal and broken diagonals, twenty times altogether in each square, or 500 times for all. Combining the five squares in col. I we have a cube in which all the 75 "straight" rows (rank, file and vertical column), all the horizontal diagonals and three of the four cubic diagonals foot up 1565. In cube III all the cubic diagonals are correct. Each cube also has seven vertical-square diagonals with the same summation. Taking together the squares in horizontal rows we find certain diagonals having the same sum, but the columns do not. The five squares in either diagonal of the large square, however, combine to produce almost perfect cubes, with rank, file, column and cubic diagonals all correct, and many diagonals of vertical squares. A still more remarkable fact is that the squares in the broken diagonals running in either direction also combine to produce cubes as nearly perfect as those first considered. Indeed, the great square seems to be an enlarged copy of the small squares, and where the cells in the small ones unite to produce S the corresponding squares in the large figure unite to produce cubes more or less perfect. Many other combinations are discoverable, but these are sufficient to illustrate the principle, and show the interrelations of the cubes and their constituent squares. The summation 1565 occurs in this figure not less than 1400 times. The plane figure containing the five cubes (or twenty-five squares) is itself a perfect square with a summation of 7825 for every rank, file, corner or broken diagonal. Furthermore all complementary pairs are balanced about the center, as in Fig. 1. Any square group of four, nine or sixteen of the small squares is magic, and if the group of nine is taken at the center it is "perfect." It is worthy of notice that all the powers of « above the first lie in the middle rank of squares, and that all other multiples of « are grouped in regular relations in the other ranks and have the same CRITICISMS AND DISCUSSIONS. 313 grouping in all the squares of any given rank. The same is true of the figure illustrating 7*, which is to be considered next. 3. Let n=7, then 8=8407. — This is so similar in all its prop- erties to the 5-construction just discussed that it hardly needs sep- arate description. It is more nearly perfect in all its parts than the 5*, having a larger proportion of its vertical-square diagonals cor- rect. Any square group of four, nine, sixteen, twenty-five or thirty- six small squares is magic, and if the group of nine or twenty-five I 255 254 4 248 10 11 245 240 18 19 237 25 231 230 28 252 6 7 249 13 243 242 16 21 235 234 24 228 30 31 225 8 250 251 5 241 15 14 244 233 23 22 236 32 226 227 29 253 3 2 256 12 246 247 9 20 238 239 17 229 27 26 232 224 34 35 221 4i 215 214 44 49 207 206 52 200 58 59 197 37 219 218 40 212 46 47 209 204 54 55 201 61 195 194 64 217 39 38 220 48 210 211 45 56 202 203 53 193 63 62 196 36 222 223 33 213 43 42 216 205 51 50 208 60 198 199 57 192 66 67 189 73 183 182 76 81 175 174 84 168 90 91 165 69 187 186 72 180 78 79 177 172 86 87 169 93 163 162 96 185 71 70 188 80 178 179 77 88 170 171 85 161 95 94 164 68 190 191 65 181 75 74 184 173 83 82 176 92 166 167 89 97 159 158 100 152 106 107 149 144 114 115 141 121 135 134 124 156 102 103 153 109 147 146 112 «7 139 138 120 132 126 127 129 104 154 155 101 145 iti 1 10 148 137 119 118 140 128 130 131 125 157 99 98 160 108 150 151 105 116 142 143] 113 133 123 122 136 Fig. 3- (4 4 ) be taken at the center of the figure it is "perfect." The grouping of multiples and powers of n is very similar to that already described for 5*. II. When n is even. I. Let «=4, then 8=514. — The numbers may be arranged in either of two ways. If we take the diagram for the 4-cube as II III 1 1295 1294 3 1292 6 1278 20 21 1276 23 "73 37 1259 1258 39 1256 42 1290 8 ■288 1287 11 7 25 1271 27 28 1268 1272 '254 44 .1252 1251 47 43 1284 1283 '5 16 >4 1279 31 32 1264 1263 1265 36 1248 1247 5' 52 50 '243 '3 '7 1281 1282 1280 18 1266 1262 34 33 35 1261 49 53 1245 1246 1244 54 12 1286 9 10 1289 1285 1267 29 1270 1269 26 30 48 1250 45 46 1253 1249 1291 3 4 '293 5 1296 24 1277 ■275 22 •274 '9 1255 38 40 '257 4' 1260 1188 no in 1 186 "3 1 183 127 1 169 1168 129 1166 '32 "52 146 '47 "50 '49 "47 "5 1 181 "7 118 1 178 1 182 1 164 134 1 162 1 161 '37 '33 '5' "45 '53 154 1142 "46 121 122 1 174 "73 "75 126 1158 "67 141 142 140 "53 '57 ■58 1 138 "37 "39 162 1176 1 172 124 123 125 1171 ■39 '43 "55 "56 "54 144 1 140 "36 160 '59 161 "35 1177 119 1180 1 179 116 120 '38 u6o '35 •36 "63 "59 1 141 '55 "44 "43 152 •56 114 1187 1185 112 1184 I09 1165 128 130 1167 '3' 1 170 '5° "5' "49 148 1 148 '45 217 1079 1078 219 1076 222 1062 236 237 1060 239 1057 253 1043 1042 255 1040 258 1074 224 1072 1071 227 223 241 i°55 243 244 1052 1056 1038 260 1036 1035 263 259 1068 1067 231 232 230 1063 247 248 1048 1047 1049 252 1032 1031 267 268 266 1027 229 233 1065 1066 1064 234 1050 1046 250 249 251 '045 265 269 1029 1030 1028 270 228 1070 225 226 1073 I069 105 1 245 1054 ">53 242 246 264 1034 261 262 1037 >°33 1075 218 220 1077 221 1080 240 106 1 1059 238 1058 235 1039 254 256 1041 257 1044 865 43t 43° 867 428 870 4H 884 885 4U 887 409 901 395 394 903 392 906 426 872 424 423 875 871 889 407 891 892 404 408 390 908 388 387 9" 907 420 419 879 880 878 415 895 896 400 399 401 900 384 &S 915 916 914 379 877 881 417 418 416 882 402 398 898 897 899 397 9'3 9'7 381 382 380 918 876 422 873 874 425 421 403 '893 406 405 890 894 912 386 909 910 389 385 427 866 868 429 869 432 888 4'3 411 886 410 883 39'- 902 904 393 9°5 396 864 434 435 862 437 859 451 845 844 453 842 456 828 470 47' 826 473 823 439 857 441 442 854 858 840 458 838 837 461 457 475 821 477 478 818 822 445 446 850 849 8S« 450 834 833 465 466 464 829 481 482 814 8>3 815 486 852 848 448 447 449 847 463 467 83' 832 830 468 816 812 484 483 485 811 853 443 856 855 440 444 462 836 459 460 839 835 8i7 479 820 819 476 480 438 863 861 436 860 433 841 452 454 843 455 846 474 827 825 472 824 469 756 542 543 754 545 751 559 737 736 56i 734 564 720 578 579 7'8 58i 7'5 547 749 549 55° 746 750 732 566 73° 729 569 565 583 7'3 585 586 710 7'4 553 554 742 741 743 558 726 725 573 574 572 721 589 59° 706 7°5 707 594 744 740 556 555 557 739 57" 575 723 724 722 576 708 704 592 59' 593 703 745 55" 748 747 548 552 57° 728 567 568 73" 727 709 587 712 7" 584 588 546 755- 753 544 752 541 733 560 562 735 563 738 582 7W 7'7 580 7J6 577 Fig. 4, First Part. (6*: 8=3891) IV VI 122$ 71 70 1227 68 1230 1224 74 75 1222 77 1219 120^ 92 93 1204 95 1201 r>r> 1232 64 f>3 '235 ■23' 79 1217 81 82 1214 1218 97 "99 99 100 119C 1200 fio 59 '239 1240 1238 55 85 86 1210 1209 [211 90 >°3 104 1192 1191 "93 108 '237 124! 57 58 56 1242 1212 1208 88 87 89 1207 "94 1190 106 ■°5 107 1189 1236 62 <m 1234 65 61 1213 83 1216 1215 80 84 "95 101 1 198 "97 98 102 (>7 1226 1228 69 1229 72 78 1223 1221 7* 1220 73 96 1205 1203 94 1202 91 180 1 1 18 1 1 19 178 1121 '75 181 '"5 1 114 '83 1 1 12 186 '99 1097 1096 20 r IO94 204 1123 '73 1125 I 126 170 '74 11 10 188 1108 1107 KJI 187 1092 206 1090 1089 209 205 1129 1130 166 "•'5 167 "34 1 104 1103 '95 196 194 1099 1086 1085 213 214 212 1081 168 164 1132 "3' "33 ""'3 '93 '97 MOI 1 102 MOO 198 211 2'5 1083 1084 I082 216 169 1127 172 171 1124 1128 192 1 106 189 190 1 109 "05 210 1088 207 208 I09I 1087 1122 '79 '77 1120 176 1 1 17 1111 182 184 "'3 185 1 1 16 1093 200 202 1095 203 1098 1009 287 286 1011 284 1014 ico8 290 291 1006 293 1003 990 308 309 988 3" 985 282 1016 280 279 1019 1015 295 IOOI 297 298 998 1002 3'3 983 3'5 3'6 980 984 276 275 1023 1024 1022 271 30' 302 994 993 995 306 3'9 320 976 975 977 324 102 1 1025 273 274 272 1026 996 992 3°4 303 305 99' 978 974 322 321 323 973 1020 278 1017 1018 281 '77 997 299 1000 999 296 300 979 3'7 982 981 3'4 3'8 283 1010 1012 285 1013 288 294 1007 1005 292 1004 289 312 989 987 3'0 986 3°7 S<" 935 934 363 932 366 360 938 939 358 94' 355 342 956 957 340 959 337 93° 368 928 927 371 367 943 353 945 946 350 354 961 335 963 964 332 336 924 923 375 376 374 919 949 95° 346 345 347 954 967 968 328 327 329 972 373 377 921 922 920 378 348 344 952 95' 953 343 330 326 970 969 97' 325 372 926 369 370 929 925 349 947 352 35' 944 948 33' 965 334 333 962 966 93' 362 3 6 4 933 365 936 942 359 357 940 356 937 960 34' 339 958 338 955 504 794 795 502 797 499 505 79' 790 5°7 788 510 523 773 772 525 77° 528 799 497 801 802 494 498 786 5'2 784 783 5'5 5" 768 53o 766 765 533 529 805 806 490 489 491 810 780 779 5'9 520 5'8 775 762 76i 537 538 536 757 492 488 808 807 809 487 5'7 52' 777 778 776 522 535 539 759 760 758 540 493 803 496 495 800 804 5'6 782 5'3 5'4 785 781 534 764 53' 532 767 763 798 503 5°' 70 500 79i 787 506 508 789 509 792 769 524 526 771 527 774 612 686 687 610 689 607 613 683 682 6'5 680 618 631 66 S 664 633 662 636 691 605 693 694 602 606 678 620 676 675 623 619 660 638 658 657 641 637 697 698 598 597 599 702 672 671 627 628 626 667 654 653 645 646 644 649 £00 595 700 699 701 595 625 629 669 670 668 630 643 647 651 652 650 648 601 695 604 603 692 696 624 674 621 622 677 673 642 656 639 640 659 655 690 611 609 688 608 685 679 614 616 681 617 684 661 632 634 663 635 666 Fig. 4, Second Part. (6':S=389i) 316 THE MONIST. given in Magic Squares and Cubes and simply extend it to cover the larger numbers involved we shall have a group of four cubes in which all the "straight" dimensions have S=SI4, but no diag- onals except the four cubic diagonals. Each horizontal row of squares will produce a cube having exactly the same properties as those in the four vertical rows. If the four squares in either diag- I 409S 4094 4 5 4091 4090 8 4032 66 67 4029 4028 70 71 4025 4088 10 11 4085 4084 14 IS 4081 73 4023 4022 76 77 4019 4018 80 4080 18 19 4077 4076 22 23 4073 81 4015 4014 84 85 401 1 4010 88 25 4071 4070 28 29 4067 4066 32 4008 90 91 4005 4004 94 95 4001 406S 31 30 4068 4069 27 26 4072 96 4002 4003 93 92 4006 4007 89 24 4074 4075 31 20 4078 4079 17 4009 87 86 4012 4013 83 82 4016 16 4082 4083 13 12 4086 4087 9 4017 79 78 4020 4021 75 74 4024 4089 7 6 4092 4093 3 2 4096 72 4026 4027 69 68 4030 4031 65 4064 34 35 4061 4060 38 39 4057 97 3999 3098 100 101 399S 3994 104 41 4055 4054 44 45 4051 4050 48 3992 106 107 3989 3988 1 10 III 3985 49 4047 4046 52 53 4043 4042 56 3984 114 115 398i 3980 118 119 3977 4040 58 59 4037 4036 62 63 4033 121 3975 3974 124 125 3971 3970 128 64 4034 4035 61 60 4038 4039 57 3969 127 126 3972 3973 123 122 3976 4041 55 54 4044 4045 51 50 4048 120 3978 3979 117 116 3982 3983 113 4049 47 46 4052 4053 43 42 4056 112 3986 3987 109 108 3990 3991 ios 40 4058 4059 37 36 4062 4063 33 3993 103 ro2 3996 3997 99 98 4000 II IV Fig. 5, 8*, First Part (One cube written). onal of the figure be piled together neither vertical columns nor cubic diagonals will have the correct summation, but all the diagonals of vertical squares in either direction will. Regarding the whole group of sixteen squares as a plane square we find it magic, having the summation 2056 in every rank, file and corner diagonal, 1028 CRITICISMS AND DISCUSSIONS. 317 in each half-rank or half-file, and 514 in each quarter-rank or quarter-file. Furthermore all complementary pairs are balanced about the center. The alternative arrangement shown in Fig. 3 makes each of the small squares perfect in itself, with every rank, file and corner diag- onal footing up 514 and complementary pairs balanced about the 3968 130 131 3965 39&t 134 135 396i 193 3903 3902 196 «97 3899 3898 200 137 3959 3958 140 141 3955 3954 144 3896 202 203 3893 3892 206 207 3889 145 3051 3950 148 149 3947 3946 152 3888 210 211 3885 3884 214 215 3881 3944 154 155 3941 3940 158 IS9 3937 217 3879 3878 220 221 3875 3874 224 160 3938 3939 157 156 3942 3943 153 3873 223 222 3876 3877 219 218 3880 3945 151 150 3948 3949 147 146 3952 216 38R? 3883 213 212 3886 3887 209 3953 143 142 3956 3957 139 138 3960 208 3890 3891 205 204 3894 3895 201 136 3962 3963 133 132 3966 3967 129 3897 199 198 3900 3901 195 194 3904 161 3935 3934 164 165 3931 393° 168 3872 226 227 3869 3868 230 231 3865 3928 170 171 3925 3924 »74 175 3921 233 3863 3862 236 237 3859 38S8 240 3920 178 179 35W7 39i6 182 183 3913 241 3855 3854 244 245 3851 3850 248 185 39" 39«o 188 «89 3907 3906 192 .1848 250 251 3845 3844 254 255 3841 3905 191 190 3908 3909 187 186 3912 256 3842 3843 253 252 3846 3847 249 184 3914 3915 181 180 3918 3919 177 3849 247 246 3852 3853 243 242 3856 176 3922 3923 173 172 3926 3927 169 3857 239 238 3860 3861 335 234 3864 3929 167 166 3932 3933 163 162 3936 232 3866 3867 229 228 3870 3871 225 VI VIII Fig. 5, 8*, Second Part (One cube written). center. As in the other arrangement the squares in each vertical or horizontal row combine to make cubes whose "straight" dimen- sions all have the right summation. In addition the new form has the two plane diagonals of each original square (eight for each cube), but sacrifices the four cubic diagonals in each cube. In lieu 318 THE MONIST. of these we find a complete set of "bent diagonals" ("Franklin") like those described for the magic cube of six in The Monist for July, 1909. If the four squares in either diagonal of the large figure be piled up it will be found that neither cubic diagonal nor vertical column is correct, but that all diagonals of vertical squares facing toward front or back are. Taken as a plane figure the whole group makes up a magic square of 16 with the summation 2056 in every rank, file or corner diagonal, half that summation in half of each of those dimensions, and one-fourth of it in each quarter dimension. 2. Let n=6, then 8=3891. — With the natural series 1. . .1296 squares were constructed which combined to produce the six magic cubes of six indicated by the Roman numerals in Fig. 4. These have all the characteristics of the 6-cube described in The Monist of July last — 108 "straight" rows, 12 plane diagonals and 24 "bent" diagonals in each cube, with the addition of 32 vertical-square diagonals if the squares are piled in a certain order. A seventh cube with the same features is made by combining the squares in the lowest horizontal row — i. e., the bottom squares of the num- bered cubes. The feature of the cubic bent diagonals is found on combining any three of the small squares, no matter in what order they are taken. In view of the recent discussion of this cube it seems unnecessary to give any further account of it now. The whole figure, made up as it is of thirty-six magic squares, is itself a magic square of 36 with the proper summation (23346) for every rank, file and corner diagonal, and the corresponding fractional part of that for each half, third or sixth of those dimen- sions. Any square group of four, nine, sixteen or twenty-five of the small squares will be magic in all its dimensions. 3. Let n=8, then 8=16388. — The numbers 1...4096 may be arranged in several different ways. If the diagrams in Mr. An- drews's book be adopted we have a group of eight cubes in which rank, file, column and cubic diagonal are correct (and in which the halves of these dimensions have the half summation), but all plane diagonals are irregular. If the plan be adopted of construct- ing the small squares of complementary couplets, as in the 6-cube, the plane diagonals are equalized at the cost of certain other features. I have used therefore a plan which combines to some extent the ad- vantages of both the others. It will be noticed that each of the small squares in Fig. 5 is CRITICISMS AND DISCUSSIONS. 319 perfect in that it has the summation 16388 for rank, file and corner diagonal (also for broken diagonals if each of the separated parts contain two, four or six — not an odd number of cells) , and in balan- cing complementary couplets. When the eight squares are piled one upon the other a cube results in which rank, file, column, the plane diagonals of each horizontal square, the four ordinary cubic diagonals and 32 cubic bent diagonals all have S=i6388. What is still more remarkable, the half of each of the "straight" dimensions and of each cubic diagonal has half that sum. Indeed this cube of eight can be sliced into eight cubes of 4 in each of which every rank, file, column and cubic diagonal has the footing 8194; and each of these 4-cubes can be subdivided into eight tiny 2-cubes in each of which the eight numbers foot up 16388. So much for the features of the single cube here presented. As a matter of fact only the one cube has actually been written out. The plan of its construction, however, is so simple and the relations of numbers so uniform in the powers of 8 that it was easy to in- vestigate the properties of the whole 8 4 scheme without having the squares actually before me. I give here the initial number of each of the eight squares in each of the eight cubes, leaving it for some one possessed of more leisure to write them all out and verify my statements as to the intercubical features. It should be remembered that in each square the number diagonally opposite the one here given is its complement, i. e., the number which added to it will give the sum 4097. I II III IV V VI VII VIII I 3840 3584 769 3072 1281 1537 2304 4064 289 545 3296 1057 2784 2528 1825 4032 321 577 3264 1089 2752 2496 1857 97 3744 3488 865 2976 1377 1633 2208 3968 385 641 3200 1 153 2688 2432 1921 161 3680 3424 929 2912 1441 1697 2144 193 3648 3392 961 2880 1473 1729 2112 3872 481 737 3104 1249 2592 2336 2017 16388 16388 16388 16388 16388 16388 16388 16388 Each of the sixty-four numbers given above will be at the upper left-hand corner of a square and its complement at the lower right-hand corner. The footings given are for these initial numbers, 320 THE MONIST. but the arrangement of numbers in the squares is such that the footing will be the same for every one of the sixty-four columns in each cube. If the numbers in each horizontal line of the table above be added they will be found to have the same sum: conse- quently the squares headed by them must make a cube as nearly perfect as the example given in Fig. 5, which is cube I of the table above. But the sum of half the numbers in each line is half of 16388, and hence each of the eight cubes formed by taking the squares in the horizontal rows is capable of subdivision into 4-cubes and 2-cubes, like our original cube. We thus have sixteen cubes, each with the characteristics described for the one presented in Fig- 5- If we pile the squares lying in the diagonal of our great square (starting with 1, 289, etc., or 2304, 2528, etc.) we find that its col- umns and cubic diagonals are not correct; but all the diagonals of its vertical squares are so, and even here the remarkable feature of the half-dimension persists. Of course there is nothing to prevent one's going still further and examining constructions involving the fifth or even higher pow- ers, but the utility of such research may well be doubted. The purpose of this article is to suggest in sketch rather than to discuss exhaus- tively an interesting field of study for some one who may have time to develop it. H. M. Kingery. Wabash College.